demonstration would be exactly the same as we now have it. But the ancient geometers, when they observed this enunciation could be made shorter, by giving a name to the ratio which the first straight line has to the last, by which name the intermediate ratios might likewise be signified, of the first to the second, and of the second to the third, and so on, if there were more of them, they called this ratio of the first to the last the ratio compounded of the ratios of the first to the second, and of the second to the third straight line: that is, in the present example, of the ratios which are the same with the ratios of the sides, and by this they expressed the proposition more briefly thus: if there be two equiangular parallelograms, they have to one another the ratio which is the same with that which is compounded of ratios that are the same with the ratios of the sides. Which is shorter than the preceding enunciation, but has precisely the same meaning. Or yet shorter thus: equiangular parallelograms have to one another the ratio which is the same with that which is compounded of the ratios of their sides. And these two enunciations, the first especially, agree to the demonstration which is now in the Greek. The proposition may be more briefly demonstrated, as Candalla does, thus: let ABCD, CEFG, be two equiangular parallelograms, and complete the parallelogram CDHG, then, because there are three parallelograms AC, CH, CF, the first AC (by the definition of compound ratio) has to the third CF, ratio which is compounded of the ratio of the first AC to the second CH, and of the ratio of CH to the third CF; but the parallelogram AC is to the parallelogram CH, as the straight line BC to B CG; and the parallelogram CH is to CF, as the straight line CD is to CE: therefore the parallelogram AC has to

A

the

D

H

G

E F

CF the ratio which is compounded of ratios that are the same with the ratios of the sides. And to this demonstration agrees the enunciation which is at present in the text, viz. equiangular parallelograms have to one another the ratio which is compounded of the ratios of the sides; for the vulgar reading, "which is compounded of their sides," is absurd. But, in this edition, we have kept the demonstration which is in the Greek text, though not so short as Candalla's; because the way of finding the ratio which is compounded of the ratios of the sides, that is, of finding the ratio of the parallelograms, is shown in that, but not in Candalla's demonstration; whereby beginners may learn, in

like cases, how to find the ratio which is compounded of two or more given ratios.

From what has been said, it may be observed, that in any magnitudes whatever of the same kind A, B, C, D, &c. the ratio compounded of the ratios of the first to the second, of the second to the third, and so on to the last, is only a name or expression, by which the ratio which the first A has to the last D is signified, and by which at the same time the ratios of all the magnitudes A to B, B to C, C to D, from the first to the last, to one another, whether they be the same, or be not the same, are indicated; as in magnitudes which are continual proportionals A, B, C, D, &c. the duplicate ratio of the first to the second is only a name or expression by which the ratio of the first A to the third C is signified, and by which, at the same time, is shown that there are two ratios of the magnitudes, from the first to the last, viz. of the first A to the second B, and of the second B to the third or last C, which are the same with one another; and the triplicate ratio of the first to the second is a name or expression by which the ratio of the first A to the fourth D is signified, and by which, at the same time, is shown that there are three ratios of the magnitudes, from the first to the last, viz. of the first A to the second B, and of B to the third C, and of C to the fourth or last D, which are all the same with one another; and so in the case of any other multiplicate ratios. And that this is the right explication of the meaning of these ratios is plain from the definitions of duplicate and triplicate ratio, in which Euclid makes use of the word ayera, is said to be, or is called; which word, he, no doubt, made use of also in the definition of compound ratio, which Theon, or some other, has expunged from the Elements; for the very same word is still retained in the wrong definition of compound ratio, which is now the 5th of the 6th book: but in the citation of these definitions it is sometimes retained, as in the demonstration of prop. 19. book 6. "the first is said to have, ze hɛɛzai, to the third the duplicate ratio," &c. which is wrong translated by Commandine and others, "has" instead of "is said to have:" and sometimes it is left out, as in the demonstration of 33. prop. of the 11th book, in which we find "the first has, z, to the third the triplicate ratio;" but without doubt εx, "has," in this place signifies the same as exew heyetat, is said to have: so likewise in prop. 23. B. 6. we find this citation, "but the ratio of K to M is compounded, vyzeral of the ratio of K to L, and the ratio of L to M," which is a shorter way of

expressing the same thing, which, according to the definition, ought to have been expressed by συγκείσθαι λέγεται, is said to be compounded.

From these remarks, together with the proposition subjoined to the 5th book, all that is found concerning compound ratio, either in the ancient or modern geometers, may be understood and explained.

PROP. XXIV. B. VI.

It seems that some unskilful editor has made up this demonstration as we now have it, out of two others; one of which may be made from the 2d prop. and the other from the 4th of this book for after he has, from the 2d of this book, and composition and permutation, demonstrated, that the sides about the angle common to the two parallelograms are proportionals, he might have immediately concluded, that the sides about the other equal angles were proportionals, viz. from prop. 34. B. 1. and prop. 7. book 5. This he does not, but proceeds to show, that the triangles and parallelograms are equiangular; and in a tedious way by help of prop. 4. of this book, and the 22d of book 5. deduces the same conclusion: from which it is plain that this ill composed demonstration is not Euclid's: these superfluous things are now left out, and a more simple demonstration is giyen from the 4th prop. of this book, the same which is in the translation from the Arabic, by help of the 2d prop. and compo sition but in this the author neglects permutation, and does not show the parallelograms to be equiangular, as is proper to do for the sake of beginners.

PROP. XXV. B. VI.

It is very evident that the demonstration which Euclid had given of this proposition has been vitiated by some unskilful hand: for, after this editor had demonstrated that "as the rectilineal figure ABC is to the rectilineal KGH, so is the parallelogram BE to the parallelogram EF;" nothing more should have been added but this, "and the rectilineal figure ABC is equal to the parallelogram BE: therefore the rectilineal KGH is equal to the parallelogram EF," viz. from prop. 14. book 5. But betwixt these two sentences he has inserted this; "wherefore, by permutation, as the rectilineal figure ABC to the parallelogram BE, so is the rectilineal KGH to the parallelogram EF;" by which

it is plain, he thought it was not evident to conclude, that the second of four proportionals is equal to the fourth from the equality of the first and third, which is a thing demonstrated in the 14th prop. of B. 5. as to conclude that the third is equal to the fourth, from the equality of the first and second, which is no where demonstrated in the Elements as we now have them: but though this proposition, viz. the third of four proportionals is equal to the fourth, if the first be equal to the second, had been given in the Elements by Euclid, as very probably it was, yet he would not have made use of it in this place; because, as was said, the conclusion could have been immediately deduced without this superfluous step, by permutation: this we have shown at the greater length, both because it affords a certain proof of the vitiation of the text of Euclid; for the very same blunder is found twice in the Greek text of prop. 23. book 11. and twice in prop. 2. B. 12. and in the 5. 11. 12. and 18th of that book; in which places of book 12. except the last of them, it is rightly left out in the Oxford edition of Commandine's translation; and also that geometers may beware of making use of permutation in the like cases for the moderns not unfrequently commit this mistake, and among others Commandine himself, in his commentary on prop. 5. book 3. p. 6. b. of Pappus Alexandrinus, and in other places the vulgar notion of proportionals has, it seems, pre-occupied many so much, that they do not sufficiently understand the true nature of them.

Besides, though the rectilineal figure ABC, to which another is to be made similar, may be of any kind whatever; yet in the demonstration the Greek text has "triangle" instead of "rectilineal figure," which error is corrected in the above named Oxford edition.

PROP. XXVII. B. VI.

The second case of this has aws, otherwise, prefixed to it, as if it was a different demonstration, which probably has been done by some unskilful librarian. Dr. Gregory has rightly left it out: the scheme of this second case ought to be marked with the same letters of the alphabet which are in the scheme of the first, as is now done.

PROP. XXVIII. and XXIX. B. VI.

These two problems, to the first of which the 27th prop. is necessary, are the most general and useful of all in the Elements,

and are most frequently made use of by the ancient geometers in the solution of other problems; and therefore are very ignorantly left out by Tacquet and Dechales in their editions of the Elements, who pretend that they are scarce of any use. 'The cases of these problems, wherein it is required to apply a rectangle which shall be equal to a given square, to a given straight line, either deficient or exceeding by a square; as also to apply a rectangle which shall be equal to another given, to a given straight line, deficient or exceeding by a square, are very often made use of by geometers. And, on this account, it is thought proper, for the sake of beginners, to give their constructions as follows:

1. To apply a rectangle which shall be equal to a given square, to a given straight line, deficient by a square; but the given square must not be greater than that upon the half of the given line.

Let AB be the given straight line, and let the square upon the given straight line C be that to which the rectangle to be applied must be equal, and this square, by the determination, is not greater than that upon half of the straight line AB.

L

A

H

F

K

B

G

Bisect AB in D, and if the square upon AD be equal to the square upon C, the thing required is done: but if it be not equal to it, AD must be greater than C, according to the determination; draw DE at right angles to AB, and make it equal to C: produce ED to F, so that EF be equal to AD or DB, and from the centre E, at the distance EF, describe a circle meeting AB in G, and upon GB describe the

C

E

כן

square GBKH, and complete the rectangle AGHL; also join EG. And because AB is bisected in D, the rectangle AG, GB together with the square of DG is equal (5. 2.) to (the square of DB, that is, of EF or EG, that is, to) the squares of ED, DG : take away the square of DG from each of these equals; therefore the remaining rectangle AG, GB is equal to the square of ED, that is, of C; but the rectangle AG, GB is the rectangle AH, because GH is equal to GB; therefore the rectangle AH is equal to the given square upon the straight line C. Wherefore the rectangle AH, equal to the given square upon C, has been